Problem: Simplify and expand the following expression: $ \dfrac{5}{5y - 5}+ \dfrac{2}{5y + 20}+ \dfrac{3y}{y^2 + 3y - 4} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{5}{5y - 5} = \dfrac{5}{5(y - 1)}$ We can factor a $5$ out of denominator in the second term: $ \dfrac{2}{5y + 20} = \dfrac{2}{5(y + 4)}$ We can factor the quadratic in the third term: $ \dfrac{3y}{y^2 + 3y - 4} = \dfrac{3y}{(y - 1)(y + 4)}$ Now we have: $ \dfrac{5}{5(y - 1)}+ \dfrac{2}{5(y + 4)}+ \dfrac{3y}{(y - 1)(y + 4)} $ The least common multiple of the denominators is: $ 25(y - 1)(y + 4)$ In order to get the first term over $25(y - 1)(y + 4)$ , multiply by $\dfrac{5(y + 4)}{5(y + 4)}$ $ \dfrac{5}{5(y - 1)} \times \dfrac{5(y + 4)}{5(y + 4)} = \dfrac{25(y + 4)}{25(y - 1)(y + 4)} $ In order to get the second term over $25(y - 1)(y + 4)$ , multiply by $\dfrac{5(y - 1)}{5(y - 1)}$ $ \dfrac{2}{5(y + 4)} \times \dfrac{5(y - 1)}{5(y - 1)} = \dfrac{10(y - 1)}{25(y - 1)(y + 4)} $ In order to get the third term over $25(y - 1)(y + 4)$ , multiply by $\dfrac{25}{25}$ $ \dfrac{3y}{(y - 1)(y + 4)} \times \dfrac{25}{25} = \dfrac{75y}{25(y - 1)(y + 4)} $ Now we have: $ \dfrac{25(y + 4)}{25(y - 1)(y + 4)} + \dfrac{10(y - 1)}{25(y - 1)(y + 4)} + \dfrac{75y}{25(y - 1)(y + 4)} $ $ = \dfrac{ 25(y + 4) + 10(y - 1) + 75y} {25(y - 1)(y + 4)} $ Expand: $ = \dfrac{25y + 100 + 10y - 10 + 75y}{25y^2 + 75y - 100} $ $ = \dfrac{110y + 90}{25y^2 + 75y - 100}$ Simplify: $ = \dfrac{22y + 18}{5y^2 + 15y - 20}$